Unsymmetrical electric wave filter



July 21, 1936. w. cAuER 2,048,426

UNSYMMETRICAL ELECTRIC WAVE FILTER Filed Nov. '23, 1955 M M f 35 @0 9 M f Hh;

@42M I I Patented July 21, 1,936

PATENT o1-FICE g lUNsinviME'rinoAL ELECTRIC WAVE FILTER`4 Y Wilhelm Cauer, v(Siotti-ngen, Germany Appliation November 23, 1933, serial No.j'699,391

In Germany November V10, 1932,'7 i K j My invention relates toelectric, four-terminal l` networks and moreV particularly to electricwave filters adapted for the selection and propagation of electromagneticA signals of prescribed. rangesl 5 -of frequency.

Inretters Patent 1,989,545, issued` January: a9, i

1935, I havedisclosed and claimed symmetrical wave filters, as well as such unsymmetricalfllters as are equivalent toa symmetrical filter connect- 1 i ed in series with a transformen'or'with a phasecorrection' circuit, or both.

one of the objects of the present invention iste provide new and improved unsymmetrical, `four terminal networks that are not equivalent to 'a symmetrical, four-terminal network in series with an, ideal transformer.

A further object isV to provide Y unsyminetrical T wave lters of the abovefdescribed character for" which the determinant of4 the ,chariacteristic.v

matrix is a constant. Such a filter willherein after be referred to as a K-D-lter.

Another object of the invention is connected with the use, in K-D-iilters, of .T-parameters" derivable fromv known Tschebyscheff-lparameters.- "f

25, Other objects ofthe invention are to previde K-Delters `'without ideal 'transformersQf-vas well as'fi novel networks containing inductive couplings;

A further object ofthe invention is to provide unsymmetrical wave lters, and more particularly 304 v`K-D-i"ilters, having all the .advantagesi-,ofvrsylriy `V`-metrical filters. As lters'of the aboveliiescribed character, furthermoregarevof more general nature than symmetrical filters, it isupossilole,A with their aid, to solve problems that lare' 4sometimes not capable of solution with the Vuse of symmetri- 4cal networks alone.

Other and stili further Vobjects will beexplained` hereinafter, andv will be particularly ,pointed out, in the appended claims, itbeing understoodthat 40 .I intend, by suitable expressions in the claims, to specify all the novelty that theuinvention `may possess. i y The invention will now be explained in connection with the accompanying drawing,` in which Figs. 1 and 2 are diagrammatic views illustrating four-terminal, symmetrical ltersuobtained ,from

an unsymmetrical network by reflecting the `same, y -and connecting the network and its reectionjin series; Fig. 3 is a diagrammatic view of anun-: V

5,0 symmetrical, high-pass,V `Kv-Df-i'llter circuitr'- ranged according to one` form of the present", invention; Fig. `4 is a diagram showing plots orcurvesof attenuation characteristicsof the filter circuits illustrated in Fig. 3i; and 5 is a simij, 55.-.-1ar diagram' Shswins .plots @r` Qurresoficrrem spending image-impedance characteristics.

In Fig.' 3, there is shown a four-terminal network, the pair of input terminalsbeing shown at l, l and the pair of output terminals at 2, 2.V

For present purposes,this network may be con- 511 sidered to represent any unsymmetrical, fourterr'ninal network, and it is proposed to show how to obtain its matrix.

nLetting Ei represent the complex voltage at the input terminals I, I; I1 .represent the'complexrl Y current at the input terminals I, l; E2 represent i the .complex voltage at the output terminals 2, -2; i and :12. represent the complex current at the'out-x: put terminals 2,72 and letting ments of the particular problem in hand. ToV

make a good approximation totzero attenuation in the transmitting bands, it is necessary that the resistances be nearly zero. In thepfollowing portion of this specification,` the resistances will be wholly neglected for this reason and also because 35 it will simplify the discussion without in reality, detracting from the generality of the method. If

the resistances then be neglected, ZnandZzz will have special values of reactances, which are char--r acteristics of two-lterminal networks. f i 40 ZnZiz I K Z12Z22 is the' characteristic` matrix of the four-terminal T45 network. Y

It may be assumed, without essentially restricting the generality oftheproblem, that a receiver and a transmitter having equal resistances R are` connected together by an electric Wave-lter. 'It 50:

will be understood, ofcourse, that the invention is not exclusively restricted to the case of equal Y resistances at the ends of the rlter, for the "caseof unequal resistancesn can be reduced to the case f of equal resistances by a serial connection of the filter with an ideal transformer, or by a network equivalent to it.

It will be convenient to deal, not with the functions Z11, Ziz and Z22 themselves, but with the reduced impedance functions 21z= R and which are also rational functions of l.

The design of constructing a filter can be always divided in two parts: first, to. obtain a suitable, realizable, characteristic matrix of the functions 211, zzz and 212; and secondly, to construct the physical network that corresponds to that matrix. The most important function for the characterization of' a filter is the working or total attenuation, represented by A. This will be defined as I denoting the'k current which would flow'in thel receiving apparatus if the sending apparatus and the receivingr alzzparatusv were immediately connected together; and vI denotingV the corresponding. current that would flow if the wave-filter were interposed between them. In'the general case,.Aiis the following function of 211, zzz anden:y

In the special case of' the K-D-lters, characterized by Y '211212 A- ,Zizzzzl-l this Vreduces to Y 21H-222+ 2- (la) A-lnl 2z12 y Froml these equations, it follows that A= if 212:0; further, that A=0, if t and where W, represents the image impedance at the input terminals I, I', andv W2 represents the image impedance at the output terminals- 2, 2'.'

That' these two latter conditions are necessary and sufiicient for ideal transmission follows* also withoutr the use of Equation (l), from the fact that av-maximum transfer of-energy from a sending apparatus withreal internal resistance and a given electromotive force to a receiving apparatus joined theretoV can take place only if this receiving resistance is equal to the internal resistance of the sendingapparatus.

It follows from the formulas y Y zu and" Y t n Zu that the condition that thel determinant of the characteristic matrix shall bea constant,

means physically that the left-side, image impedance, or the surge impedance, W1, is R2 times the reciprocal of the right-side image impedance W2:

as appears from Fig. 5. 'I'his will hereinafter be referred to as the constant determinant condition. f

As an ideal transmission in the complete frequency ranges is required, in the practical case,

f networks having reactances only are applicable.

rAs zu yand zzz arev reactances, they can have only discrete poles alternating always with the zeros. The requirement of a very high attenuation in a complete frequency. range can only be fulfilled, therefore, by having `from which vit follows that this requirement is intheattenuaton rangesof F, that the left half of the filter; alone attenuates.

The complete symmetrical lter, willY therefore do the same. y

Furthermore, the decisive function of the four-terminall attenuation or attenuation constant A1 Vof a symmetrical filter, being approximatelyfV l` in the attenuation ranges, will be obtainedby 2) D: from which it follows that isthe same thing ashaving v D=1 On the other, hand, W1A is the image impedance of the symmetrical filter illustrated in Fig. 1; that is, it is the function in symmetrical, wave filters; and W2 is the image impedance of symmetrical, four-terminal networksillustrated in Fig. 2, obtained, similarly, from reiection of the filter F on the other side, and arrangement in series. It therefore follows fromthe theory of symmetrical filters, as de- Y scribed'n the aforesaid Letters Patent, that, ac-

cordingto the type' of filter (whether low-pass, lfiighpass, band-pass and band-elimination or low-and-high-pass filter), only certain classes of functions of the frequency parameter A can be used,V` But the selection of these classes, and of the parameters in the different classes, is no more arbitrary than it is in the case of the symmetrical filters, sincerthequestion is now of symmetricalY two-terminal reactances.

filters of the special structure illustrated in Fig. 1 or Fig. 2.

To restrict further the classes and parameters, one may employ my theorem concerning the necessary and sufficient conditions for the realization of any arbitrary unsymmetrical, reactance, four-terminal network (see Reaktanz theorem of W. Cauer, Sitzungsberichte der preussischen Akademie der Wissenschaften, Berlin, `1931). This theorem reads:

In order that a'quadratid symmetrical matrix 'ZOO (Zst) with no rowsmay be assumed as the character. istic matrix of `a reactance, 2n-terminalnetwork, it is necessarythat the associate quadratic Yform Il i 22st 1,1,

for each real `systemof values Y 11,12 ,Inl

shall satisfy the conditions which are true for Elements of such matn rices, and of such matrices only, can be written in the form f The same conditions are sufficient if n l ideal transformers are added to the -reactances In the present case, n is to be chosen equal to 2. The construction of general, unsymmetrical wave filters follow readily as aV result of these Conditions. `W1, vW2 and D maybe arbitrarily selected-according-tothe type and the cut-01T (limiting) frequencies of the Ifilter-from the known classes for Y(ZT.

and Y N/ZL Then Zu results las W1D,-Z22 as W2D (both of the realizable reactances) and R2A=Z11Z22Zi22 as W1W2. It is next in order to find out if yz,2=/W1W2(D2- 1) is not contradictory to the above ,conditions of realization, `and the choice `of the classes and parameters has to be undertaken -with this in mind.

Of the numerousnew unsymmetrical -wave- `Because filters thus obtainable,V it is desired to make special note of the K-D-lters, where A numerical, ,concrete example will make this more clear. I-Iere,

Z12= R1/ 13 2-T.

Condition (4) is, therefore, satisfied; else A could not be independent of x. The classes of functions of the following Table I for are then possible. The invention `is applicable to other filters of the LP, HP, BP and BE type, as Well as to `filters of higher types, They maybe derived by .the rules hereinafter given immediately after the formulas for W1 for the bandelimination filters (BE).

TABILEyI Band-eZmtnation-jlters (BE) and a rejector frequency of w3 is a frequency for JW 1 which w/LI-wi If one chooses Y v+m: w1=w2,

w/zi-w-lzw/V--wi -wa obtains double rejector frequencies. By `such M -1502+@ 2) combinations, such D-classes (see Table 1) can (3) HIT--:L-z be found with such parameters that all rejector Fw-1 ).1/ +91 Y frequencies are double and (4) (zlw-taOJIwan) A 1/)62 -lw 12(}\2l wh/A2 w12 212 :1/ 211222--1 2 2 2 2 2 2 becomes rational and real. Y (5) mwwf/? Good filters have the quality that their resow' wb @L nance and anti-resonance-frequencies are closer (6) (V+w-2)(2lwo2)(7\2lwa2) and `closer together as theV limiting frequencies Yare approached. A specialchoice of such para- The parameters of. the Dvi-classes oughtito be determined in such a manner that Zig becomes real and rational.` The followingpmethod for attaining this result presents itself. The general rule is true that if w1 and wz are possible of the same type (for example, LP) of symmetri-i.`

cal filters, this is true alsolfor (6) W3- Wi+W2 y Then wa (as Well as a rejector frequency of 102 is `afreql'ien'cfyfor' meters which comprises, in addition to the postulate of physical realizability (that is, that Zia shall be real and' rational), even further restrictions, resulting in obtaining most valuable filters of a certain class, are the T-parameters. These parameters are nearly identical with the Tschebyscheff-parameters Ydefined in the Siebschaltungen. factor m or ,u is chosen in a different Way, so that the largest value of` in the intervals of approximation '(T-#intervals is 1. 'Ihen the smallest value in the intervals of approximation is I-I-2 or e2; or, expressed otherwise: the breadth of the region where varies in the T-intervals is 2 ln H or 2 ln 9. This is illustrated by Curve 2, Fig. 4. It thus results that all rejector frequencies are double, so that 212 becomes rational. The T-parameters can be which 192:1 found from the following Table III.

TABLE` III T-Parameters f P `Br i -V t lass Class; 2 m v. L- a b c d w/l-H* "f 1 :1i-3- im? [fabi s l .V195 1 "L4-5K- :m .fuggi sn5 k'lbd ,i A Y:

Mii-1(2)] Theonly difference is thattheconstant`V .dition that it be positive and less than 1.

In Table III, sn signifies the Jacobian elliptic 4 `function, 1c1 1, or K 1 is the modulus, and K or K is the complete elliptic integral of the first kind.

As can be seen, all parameters appearing depend on but a single independent parameter, namely, lc-l or K which has to satisfy only the single con- An interesting property of these T-parameters is that the places 9mm of the equal high minima of the attenuation, expressed by normalized frequencies, are given by the ratio of the normalized limits lc (or K) of the T-intervals and the normalized resonance frequencies (zeros) y b,d,...(or,,...)

of the attenuation function Y (or, in other words, of each of the open-circuit mpedances Zu, Z22 in the pass-band or bands). A further property of the T-parameters, which can be easilyV checked is that to each normalized anti-resonance frequency l Y @,de, (a,fy,e, .l l of the open-circuit impedances in the pass-band 45er -bands, there corresponds. a place of infinite attenuation such that the product of the corresponding (normalized) frequencies is equal-to the Y normalized frequency Ic (K) of the limit of the T- interval. V'I'his connecting law valso applies when Y rthe frequencyis zero or infinite, it being possible to associate with the. anti-resonance frequency Zero of the open circuit impedances of a low passlter, a place of infinite attenuation at a vfrequency of infinity, and conversely with' the highpass rfilter and analogously in the case of BP and'BE filters. This will be verified inthe-nlu- Vmerical example below.

The connection between the normalized frequencies and the ordinary pulsatances w is given for the different types of filters (see also Siebschaltungen Table XI) by the following formulas:l Y. for LP and HP:

for BP and BE: Y A 2 2 .4m-lealV K where Y Y j Y (wi-y-Dw A Y In the classes 3/2, y/2 and f/2, in which-only one free parameter appears,.except the limiting Y frequencies, each physically possible system of parameters is a system of T-parameters. It is to be observed, furthermore, that the parameters realized in practice may, of course, deviate some- V'what from the exact values given in Table III. In practice, the admissible tolerances affecting a modication of the computed minimum attenuation should be not more than 10 percent.

The attenuation Vconstant A1 of the K-D-filter is defined as the half of the attenuation constant of the appropriate symmetrical filter obtained by doubling (in the Siebschaltungen the same letterV A1 is used). In the case of T-parameters, the minimum value of A1 is given in the T-intervals by the formula 1n 2:0.35 Napier (see also the following numerical example). After choosing a D-class and a Wi-class with their suitable parameters for a K-D-filter, where the parameters of the Wi-class may be chosen as Tschebyscheif-parameters according to the Siebschaltungen (the application of these parameters for the image-impedance classes of the K-D-flters is a feature of the present invention), Zn,VV Z22 and Z12 are perfectly determined as the d following functions of the frequency:

In a manner analogous to the convention vadopted. in the case of symmetrical lters, the first member of a certain combination (D, W1) will represent the attenuation, and the second member the image-impedance characteristic. For example, a lter of the class y/2 2 signifies an `I-IP-iilter of constant determinant with D-characteristic fy/2 and image-impedance characteristic `2. It will be understood that a K-D-filter of a certain class with that convention and with a certain and zll

shall comprise, in this specicationas well as in the claims, not only the above-treated filters, but

also the filters produced from these K-D-filtersv 'Ihat matrix, derived in the manner just de scribed, is always physically realizable. The partial-fraction-network represented by Fig. 2b

:of the Reaktanztheorem is one kind of realization. A numerical example of filter design Will be given, using the partial-fraction-network just mentioned, involving the HP-filter of constant determinant of the class 7/2 2, shown, with lthenumerical values of its circuit elements, in Fig. 3. The following technical requirements may be prescribed: attenuation constant A1 at scheiT-:parameters for m and wa. As mentioned 10..A

before, the parameters c, wa, w are T-parameters.

According to.A Table of. the Siebschaltungen, a2=0112 corresponds to a 5 percent deviation of image-impedance in Class 2. From the third column of Table XI of the Siebschaltungen;forC1ass 2,:

`m=H= 1.051 and f a2=a2w12=28.05.104. The value of A1 min of the corresponding symmetricaliilteris-I 1 f? 2(3|1/ ln 2)='6.7 Napier, resulting from the initial requirements of Vthe problem. Corresponding to this value, f

v Y `1.154 is found in Table XIV of the Siebschaltungen?. With this value k, one'nds, from Table III cf 30 this specification, or from Table XIV of the Siebschaltungen, i Y

least three Napiers up to a frequency of 860 per 2 1 2 second, the nominal Value of the image-impeda :m :i ance in the pass region beginning with the fre- From Table 111, 35 1'" quency 1100, 500 ohms, with a 5 percent deviaz 2 2 459106 tion. The limiting frequency w1 shall be assumed w00 fx a" as 21T .1000 radian per second. As mentioned, the w2=2w1 100,8;106, parameters of class fy/2 are always T-parameters. j z- 3 From Table I, for class fy/2 u= d-2w1= 13,8.101 40 W/ x/rmmwz) A.2=39,45.1o6 A f ZZ: u 22 1002+001?) Accordingto this, the characteristic matrix From Table II,it` follows, for c1assi2 '(I-IP) 1 ZIV1 Zw) yx/)3+ w12 Z1`2Z722 7 45 m?) in partial-fraction development, becomes:

' 0,812.1om YQ sMii-ii i 10,744.1fo6iA i ;."*:00000" v+m 0,000

Tini- 000000 wwf L iufwg +0034 By multiplication and division and multiplica- Using the Reaktanztheorem, the numerical valtion with R, one obtains: j ues given in Fig. 3 will be obtained. It canbe R V+ 2 seen, from Figs. 4 and 5, that the technical re- 55` (7) Z11=$ K2 wOfi-wa): quir'ementsvofthe problem Aare fullled for the +w h (am h (o, attenuation *and the image impedance. Using 1jr ,1L--+huj i,V Fig.4,itis possimefurtherwverifythe property w of the T-parameters mentioned above by this ex- Where ample, It follows: 60* 1. mgm. rigen-warmem. f A 0 l (fw) `,c,.,',=%=o.542,o =l-=o.'sos,

h (O) Rmw2wa2..h (M njl., Y' .Y a 11 'T' waz 'y and by4 the `frequency transformation for I-IP: 2 2 2 65" Z22: R 2+Q12)(2+B22) nwo +0 )o I wa') one obtains:

Imm Lum wminL--Qminwl Where fmiiimis-f1=542 per second, fm ==803v`cycles per 70"'Y- (a) R(w12-wa2)(w2'wa2 second. bz2 -W i mpQnaz-wuz) Y R( 2)( 2 1 2) i R The miinite anti-resonance frequency, furtherh22ia w; 112200:* more, of Zn .and Z22 corresponds to an infinite f fnwi 39"" )Y I m attenuation at 1:50, 1 w 75T output terminals 2, 2', contains a condenser 2 and` The filter vof Y Fig., 3l comprises two circui branches.

I, and two coils "I, 8 in series' between the input terminals I, I. The second branch, between the a coil 3 in parallel, connected in series with two coils 4 and 5, a condenser 6 being disposed in parallel to the coil 5. The condensers and coils have the values indicated in Fig. 3. 'Ihe coil 'Iis coupled to the coil Land thefcoil 8 tothe coil 5.

Two rthings'should be noticedin connection' with the network Fig. 3. First, Vit contains no ideal transformer, though, as thev paper Reaktanztheorem demonstrates, ideall transformers cannot always be avoided in unsymmetrical, reacbe noted that there is a circuit mesh connectingY the input terminals I, I'and another circuit mesh" -connecting the output terminals 2, 2'.; that these circuits are inductively coupled together by meansof the coils, 4, 1 and 5, 8 respectively, above and tance, four-terminal networks. Secondly, it is to below an intermediate point andthat the coupling of at least one pair of coils, the upper coils. 4, I, has the opposite sign to the couplingof atA least one pair of coils, the pair of coils 8, 5 below. This opposite-sign coupling is indicated by the indication that the ratio 1:0.918 is positive and by the negative sign before the ratio 110.954, the former corresponding to the coils 8 and 5, and the latter to the coils 1 and 4. In some cases, of

course, the coils 4 and l, or 8 and 5, may be re" without further description. It is therefore desired that the appended claims be broadly'construed, unlimited except insofar as limitations may be necessary to-be imposed by the state of the prior art.`

'1. A wave-filter network comprising a first terr minal, a second terminal, a'third terminal and a fourth terminal, a circuit connecting the first and theA second terminals, a circuit connecting the third and the fourth terminals, the circuits being inductively coupled to one side and to the otherside of anl intermediate point, and an inductiveY couplingV to the one Yside of the intermediate point being negative relative to an inductive coupling to the other side of the intermediate point,

the image impedance atrthedfirst and the second terminals being the reciprocal ofthe image impedance at the third and the fourth terminals.

2. A Wave-filter according to claim 1 that is unprovided with a practically ideal transformer.

3. A wave-filter according to claim 1 in which one of the coupled pair of coils is a singlev coil commonto both circuits.

4. A wave-filter according to claim 1 in which the coupling is tight. .1..

5. A filter networknhavingjtwo pairs of terminals, the image impedance at one of the pairs of v terminals being the reciprocal of that at-the other pair'of terminals, characterizedby the fact that' the following conditions are satisfied: (a) that to each anti-resonance frequency of the open-circuit impedances in the pass-band or -bands a correspending place of infinite attenuation can Abe re- The first branch contains a condenser' Ypair of terminals, characterized by the fact that This 4checks with Fig. ll.v

lated in that part of the attenuation bander' parts ofthe attenuation bands in .which the attenuation is to correspond at least to the predetermined value (T-interval or T-intervals) such that the product of the corresponding (normal- 5 ized) frequencies is approximately equal to the limit of the T-interval expressed by normalized frequency; (b) that in the part of the pass-band or the parts of the pass-bands in which the image impedance has to have the predetermined value 10 (Tschebyschei-interval or -intervals) the minima or-maxima of the image-impedance deviate from the predetermined Value by an approximately equal amount, and that the products of the (normalized) frequencies lof these minima and maxl5 ima with lthe corresponding resonance and antiresonance frequencies of the openA circuit impedances in the attenuation-region or -regions are approximately equal to the normalized frequency of the limit of the Tschebyschei-interval.

6. A filter comprising two pairs of terminals, a reactance network with mutual inductance between these terminals, the structure of said network not being separable in several impedance branches adapted to determine the characteristics of the filter, the image impedance at one of the pair of terminals being the reciprocal of the imageimpedance at the other pair of terminals, characterized by the fact that the following conditions are satisfied: (a) that theresonance and anti-resonance frequencies of the two open-circuit impedances of the network are coincident in the pass-band or -bands and-are there closer and closer together as the limiting frequencies are approached; (b) that the resonance and anti- 35 resonance frequencies of one of the open-circuit impedances are coincident respectively with the anti-resonances and resonances of the other opencircuit impedance in the attenuation-band or -bands and are there closer and closer together as the limiting frequencies are approached.

'7. A filter network having two pairs of terminals, the image impedance at one of the pairs of terminals being the reciprocal ofjthat at the other 45 the following conditions are satisfied: (a) that the minima of the product of both the open-circuit iinpedances are approximately equally high Yin that part of the attenuation band or parts of the attenuation bands in which the attenuation is to correspond at least to the predetermined vvalue (T-interval or T-intervals) and that the places of these V,minima, expressed in terms of normalized frequency, are approximately given by the ratio of the limit of the normalized T- interval to the normalized resonance frequencies of the open-circuit impedances in the pass-band or -bands; (b) that the minima and maxima of the quotient of both the open-circuit impedances deviate with an approximately equal amount from the value one in that part of the pass-band or those parts of the pass-bands in which the image` impedances have to have the predetermined value (Tschebyscheff-inteval' or -interva1s) and that the places of said minima and maxima, expressed 65 in terms of ,normalized frequency, are approxi-` mately given by the ratio of the limit of the normalized Tschebyscheff-interval to the normalized resonance (anti-resonance) and anti-resonance (resonance) frequencies of the one (the other) 70 open-circuit impedance in the attenuation range, where the resonances of the one open-circuit impedance are the anti-resonances of the other and conversely.

8. A filter network having two pairs of terminals, the image impedance at one of the pairs of terminals being the reciprocal of that at the other pair of terminals, characterized by the fact that the following condition is satisfied: that to each anti-resonance frequency of the open-circuit impedances in the pass-band or -bands a corresponding place of infinite attenuation can be related in that part of the attenuation band or parts of the attenuation bands in which the attenuation is to correspond at least to the p redetermined value (T-interval or T-intervals), such that the product of the corresponding (normalized) frequencies is approximately equal to the limit of the T-interval expressed by normalized frequency.

9. A lter comprising two pairs of terminals, a reactance network with mutual inductance between these terminals, the structure of said network not being separable in several impedance branches adapted to determine the characteristics of the filter, the image impedance at one of the pairs of terminals being the reciprocal of the image impedance at the other pair of terminals., characterized by the fact that the following condition is satisfied: lthat the resonance and anti-resonance frequencies of the two open-cir-V cuit impedances of the network are coincident in the pass-band or -bands and are there closer and closer together as the limiting frequencies are approached. i

10. A lter comprising two pairs of terminals, a reactance network with mutual inductance between these terminals, the structure of said network not being separable in several impedance branches adapted to determine the characteristics of the filter, the image impedance at one of the pairs of terminals being the reciprocal of the image impedance at the other pair of terminals,

lcharacterized by the fact that the following condition is satisfied: that the resonance and antiresonance frequencies of one of the open-circuit impedances are coincident respectively with the attenuation bands in which the attenuation isV to correspond at least to the predetermined value (T-interval or T-intervals) and that the places of these minima, expressed in terms of normalized frequency, are approximately given by the ratio of the limit of the normalized T-interval to the normalized resonance frequencies of the open-circuit impedances in the pass-band or bands.

l2. A lter network having two pairs of terminals, the image impedance at one of the pairs of terminals being the reciprocal of that at the other pair of terminals,characterized by the fact that the following condition is satisfied: that the minima and maxima of the quotient of both the open-circuit impedances deviate with an approximately equal amount from the value one in that part of the pass-band or those parts of the passbands in which the image impedances have to have the predetermined value (Tschebyschei-interval or -intervals), and that the places of said minima and maxima, expressed in terms of normalized frequency, are approximately given by the ratio of the limit of the normalized Tschebyscheiq interval to thenormalized resonance (anti-resonance) and anti-resonance (resonance) frequencies of the one (the other) open-circuit impedance in the attenuation range, where the resonances of the one open-circuit impedance are the anti-resonances of the other and conversely.

WILHELM CAUER. 

